Question

consider the following. (if an answer does not exist, enter dne.)

$f(x)=x^{3}-6x^{2}+5$

(a) find the interval(s) of increase. (enter your answer using interval notation.)

find the interval(s) of decrease. (enter your answer using interval notation.)

(b) find the local minimum value(s). (enter your answers as a comma-separated list.)

find the local maximum value(s). (enter your answers as a comma-separated list.)

(c) find the inflection point.

$(x,y)=\left(\
ight)$

find the interval(s) where the function is concave up. (enter your answer using interval notation.)

find the interval(s) where the function is concave down. (enter your answer using interval notation.)

Explanation:

Step1: Find the first derivative

Using the power rule \((x^n)^\prime=nx^{n - 1}\), for \(y = f(x)=x^{3}-6x^{2}+5\), the first derivative \(f^\prime(x)=3x^{2}-12x=3x(x - 4)\).
Set \(f^\prime(x)=0\), then \(3x(x - 4)=0\), which gives \(x = 0\) and \(x = 4\).
We test the intervals \((-\infty,0)\), \((0,4)\) and \((4,\infty)\).

• For \(x\in(-\infty,0)\), let \(x=-1\), then \(f^\prime(-1)=3\times(-1)\times(-1 - 4)=15>0\).
• For \(x\in(0,4)\), let \(x = 1\), then \(f^\prime(1)=3\times1\times(1 - 4)=-9<0\).
• For \(x\in(4,\infty)\), let \(x = 5\), then \(f^\prime(5)=3\times5\times(5 - 4)=15>0\).

Step2: Find the second derivative

Differentiate \(f^\prime(x)=3x^{2}-12x\) using the power rule. The second derivative \(f^{\prime\prime}(x)=6x-12 = 6(x - 2)\).
Set \(f^{\prime\prime}(x)=0\), then \(6(x - 2)=0\), which gives \(x = 2\).
We test the intervals \((-\infty,2)\) and \((2,\infty)\).

• For \(x\in(-\infty,2)\), let \(x = 1\), then \(f^{\prime\prime}(1)=6\times(1 - 2)=-6<0\).
• For \(x\in(2,\infty)\), let \(x = 3\), then \(f^{\prime\prime}(3)=6\times(3 - 2)=6>0\).

Step3: Calculate function values

• Local extrema:
• When \(x = 0\), \(f(0)=0^{3}-6\times0^{2}+5 = 5\).
• When \(x = 4\), \(f(4)=4^{3}-6\times4^{2}+5=64-96 + 5=-27\).
• Inflection - point:

When \(x = 2\), \(f(2)=2^{3}-6\times2^{2}+5=8-24 + 5=-11\).

Answer:

(a)

• Interval of increase: \((-\infty,0)\cup(4,\infty)\)
• Interval of decrease: \((0,4)\)

(b)

• Local minimum value: \(-27\)
• Local maximum value: \(5\)

(c)

• Inflection point: \((2,-11)\)
• Interval where concave up: \((2,\infty)\)
• Interval where concave down: \((-\infty,2)\)